Learning by Writing – Math Writings

Integral Zariski Decomposition

Algebraic Geometry

We present a construction of integral Zariski decomposition for a pseudo-effective divisor on a normal complete algebraic surface.

Symplectic Basis of a Symplectic Vector Space

Symplectic Basis

Differential form

We construct a symplectic basis for a symplectic vector space using a method similar to the Gram-Schmidt algorithm for inner product spaces.

Pythagorean Triples - An Application of Complex Numbers

Complex Analysis

College Algebra

Solving \(a^2+b^2=c^2\) within the set of natural numbers using complex numbers.

Different Aspects of Blowing-up

Algebraic Geometry

Resolution of singularities plays an essential role in biratonal geometry. One approach is the blowing-up. In this notes, we will explore different approaches to blowing-up.

Albanese Map of a Riemannian Manifold

Differential Geometry

Algebraic Geometry

Given a compact Riemannian manifold \(M\) such that the first Betti number \(b\) is nonzero, there exists a map \(\operatorname{alb}: M\to \mathbb{R}^b/\mathbb{Z}^b\), called the Albanese map. The aim of post is to prove the smoothness of the Albanese map.

Coadjoint Orbits

Differential Geometry

Coadjoint orbits are examples of symplectic manifolds. This post aims at defining coadjoint orbits with a review on Lie group action.

A Brief Introduction to Cohomology of Lie Algebra

Differential Geometry

Let \(G\) be a compact simply connected Lie group and \(\mathfrak{g}\) its Lie algebra. In this posts, we will discuss the motivation of Lie algebra cohomology of \(\mathfrak{g}\) and its connection with the de Rham cohomology of \(G\).

Zariski Decomposition on Algebraic Surfaces

Algebraic Geometry

Zariski decompostion of pseudoeffective divisors on algebraic surface has many applications. In this post, we study Zariski decomposition on algebraic spaces of dimension 2 following an idea of Sakai.

Nakayama’s Lemma and Some Applications

Algebraic Geometry

Nakayama’s lemma is a powerful and useful tool in algebraic geometry. In this post, we will consider various versions of Nakayama’s lemma and some applications.

Seshadri Constants and Restricted Volumes

Algebraic Geometry

The purpose of this post is to show an application of the differentiation technique on multiplicity loci to Seshadri constants.

Hodge Index Theorem

Algebraic Geometry

Hodge index theorem and its variations is a fundamental tool in study of smooth projective varieties. We aim at proving Hodge index theorem on higher dimensional varieties over an algebraically closed field of arbitrary characteristic.

Poisson Bivectors

Differential Geometry

Equivalence between Poisson brackets and Poisson bivectors will be studied in this post.

Multiplicity Loci

Algebraic Geometry

Bundles of differential operators and their application to multiplicity loci will be discussed in this post.

Bertini’s Theorem

Algebraic Geometry

We present a proof of Bertini’s theorem using differentiation in parameter directions to lower the multiplicities of divisors in families.

Bundles of Principal Parts

Algebraic Geometry

Definitions and properties of bundles of principal parts (also known as jet bundles) and sheaves of differential operators will be studied in this post.